EXERCISE 8D
8 In how many ways can 5 different books be arranged on a shelf if:
a there are no restrictions
b books X and Y must be together
c books X and Y must not be together?
9 10 students sit in a row of 10 chairs. In how many ways can this be done if:
a there are no restrictions
b students A, B, and C insist on sitting together?
10 How many three-digit numbers can be made using the digits 0, 1, 3, 5, and 8 at most once each, if:
a there are no restrictions
b the numbers must be less than 500
c the numbers must be even and greater than 300?
11 Consider the letters of the word MONDAY. How many different permutations of these four letters
can be chosen if:
a there are no restrictions
b at least one vowel (A or O) must be used
c the two vowels are not together?
12 Nine boxes are each labelled with a different whole number from 1 to 9. Five people are allowed
to take one box each. In how many different ways can this be done if:
a there are no restrictions
b the first three people decide that they will take even numbered boxes?
13 Alice has booked ten adjacent front-row seats for a basketball game for herself and nine friends.
a How many different arrangements are possible if there are no restrictions?
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b Due to a severe snowstorm, only five of Alice’s friends are able to join her for the game. In
how many different ways can they be seated in the 10 seats if:
i there are no restrictions
ii any two of Alice’s friends are to sit next to her?
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EXERCISE 8E
A committee of 4 is chosen from 7 men and 6 women. How many different committees can be
chosen if:
a there are no restrictions
b there must be 2 of each sex
c there must be at least one of each sex?
a There are 7 + 6 = 13 people up for selection and we want any 4 of them.
¡ 13 ¢
= 715 possible combinations.
There are
4
¡ ¢
b The 2 men can be chosen out of 7 in 72 ways.
¡ ¢
The 2 women can be chosen out of 6 in 62 ways.
¡7¢ ¡6¢
) there are
2 £ 2 = 315 possible combinations.
c
The total number of combinations
= the number with 3 men and 1 woman + the number with 2 men and 2 women
+ the number with 1 man and 3 women
¡7¢ ¡6¢ ¡7¢ ¡6¢ ¡7¢ ¡6¢
= 3 + 1 + 2 £ 2 + 1 £ 3
= 665
or The total number of combinations
¡ ¢
¡ the number with all men ¡ the number with all women
= 13
4
¡ 13 ¢ ¡ 7 ¢ ¡ 6 ¢ ¡ 7 ¢ ¡ 6 ¢
= 4 ¡ 4 £ 0 ¡ 0 £ 4
= 665
4
a How many different committees of 3 can be selected from 13 candidates?
b How many of these committees consist of the president and 2 others?
5
a How many different teams of 5 can be selected from a squad of 12?
b How many of these teams contain:
i the captain and vice-captain
ii exactly one of the captain or the vice-captain?
6 A team of 9 is selected from a squad of 15. 3 particular players must be included, and another must
be excluded because of injury. In how many ways can the team be chosen?
7 In how many ways can 4 people be selected from 10 if:
a one person must be selected
b two people are excluded from every selection
c one person is always included and two people are always excluded?
9 A committee of 5 is chosen from 6 doctors, 3 dentists, and 7 others.
Determine the number of ways of selecting the committee if it is to contain:
a exactly 2 doctors and 1 dentist
b exactly 2 doctors
c at least one person from each of the two given professions.
15 Line A contains 10 points and line B contains 7 points.
If all points on line A are joined to all points on
line B, determine the maximum number of points of
intersection between the new lines constructed.
A
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EXERCISE 8F
8 Expand and simplify (2x + 3)(x + 1)4 .
9 Find the coefficient of:
a a3 b2 in the expansion of (3a + b)5
b a3 b3 in the expansion of (2a + 3b)6 .
EXERCISE 8G
1 Write down the first three and last two terms of the following binomial expansions. Do not simplify
your answers.
³
´
³
´
2 15
3 20
b 3x +
c 2x ¡
a (1 + 2x)11
x
x
2 Without simplifying, write down:
¡ 2
¢9
x +y
³
´
1 21
d the 9th term of 2x2 ¡
.
a the 6th term of (2x + 5)15
³
´
2 17
c the 10th term of x ¡
b the 4th term of
x
x
3 Consider the expansion of (x + b)7 .
a Write down the general term of the expansion.
b Find b given that the coefficient of x4 is ¡280.
4 Find the constant term in the expansion of:
³
´
2 15
a x+ 2
³
´
3 9
x¡ 2 .
b
x
x
6 Find the coefficient of:
a x10 in the expansion of (3 + 2x2 )10
c x6 y3 in the expansion of
7
b x3 in the expansion of
¡ 2
¢6
2x ¡ 3y
d x12
³
´
3 6
2x2 ¡
x
³
´
1 12
in the expansion of 2x2 ¡
.
x
a Find the coefficient of x5 in the expansion of (x + 2)(x2 + 1)8 .
b Find the term containing x6 in the expansion of (2 ¡ x)(3x + 1)9 . Simplify your answer.
¡
¢6
8 Consider the expression x2 y ¡ 2y2 . Find the term in which x and y are raised to the same
power.
9
a The third term of (1 + x)n is 36x2 . Find the fourth term.
b If (1 + kx)n = 1 ¡ 12x + 60x2 ¡ :::: , find the values of k and n.
³
´
1 10
is 15.
10 Find a if the coefficient of x11 in the expansion of x2 +
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REVIEW SET 8A
1 Simplify:
NON-CALCULATOR
a
n!
(n ¡ 2)!
n! + (n + 1)!
n!
b
2 Eight people enter a room and each person shakes hands with every other person. How many
hand shakes are made?
3 The letters P, Q, R, S, and T are to be arranged in a row. How many of the possible arrangements:
a end with T
4
b begin with P and end with T?
a How many three digit numbers can be formed using the digits 0 to 9?
b How many of these numbers are divisible by 5?
5 The first two terms in a binomial expansion are: (a + b)4 = e4x ¡ 4e2x + ::::
a Find a and b.
b Copy and complete the expansion.
p
p
6 Expand and simplify ( 3 + 2)5 giving your answer in the form a + b 3, a, b 2 Z .
³
´
1 8
.
7 Find the constant term in the expansion of 3x2 +
x
4
includes the term 22x3 .
8 Find c given that the expansion (1 + cx) (1 + x)
9
a Write down the first four and last two terms of the binomial expansion (2 + x)n .
b Hence find, in simplest form, the sum of the series
¡ ¢
¡ ¢
¡ ¢
2n + n1 2n¡1 + n2 2n¡2 + n3 2n¡3 + :::: + 2n + 1.
REVIEW SET 8B
CALCULATOR
1 Ten points are located on a 2-dimensional plane such that no three points are collinear.
a How many line segments joining two points can be drawn?
b How many different triangles can be drawn by connecting all 10 points with line segments
in every possible way?
2 Use the expansion of (4 + x)3 to find the exact value of (4:02)3 .
³
´
2 6
3 Find the term independent of x in the expansion of 3x ¡ 2 .
x
3
4 Find the coefficient of x
6
in the expansion of (x + 5) .
5 A team of five is chosen from six men and four women.
a How many different teams are possible with no restrictions?
b How many different teams contain at least one person of each sex?
6 A four digit number is constructed using the digits 0, 1, 2, 3, ...., 9 at most once each.
a How many numbers are possible?
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b How many of these numbers are divisible by 5?
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7 Use Pascal’s triangle to expand (a + b)6 .
a (x ¡ 3)6
b
³
´12
3
in the expansion of 2x ¡ 2
.
Hence, find the binomial expansion of:
8 Find the coefficient of
x¡6
³
´
1 6
1+
x
x
9 Find the coefficient of x5 in the expansion of (2x + 3)(x ¡ 2)6 .
³
´
1 9
is 288.
10 Find the possible values of a if the coefficient of x3 in 2x + 2
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REVIEW SET 8C
1 Alpha-numeric number plates have two letters followed by four digits. How many plates are
possible if:
a there are no restrictions
c no letter or digit may be repeated?
2
b the first letter must be a vowel
a How many committees of five can be selected from eight men and seven women?
b How many of the committees contain two men and three women?
c How many of the committees contain at least one man?
a (x ¡ 2y)3
3 Use the binomial expansion to find:
b (3x + 2)4
4 Find the coefficient of x3 in the expansion of (2x + 5)6 .
³
´
1 6
.
5 Find the constant term in the expansion of 2x2 ¡
x
6 The first three terms in the expansion of (1 + kx)
n
are 1 ¡ 4x +
15 2
2 x .
Find k and n.
7 Eight people enter a room and sit at random in a row of eight chairs. In how many ways can
the sisters Cathy, Robyn, and Jane sit together in the row?
8 A team of eight is chosen from 11 men and 7 women. How many different teams are possible
if there:
a are no restrictions
b must be four of each sex on the team
c must be at least two women on the team?
9 Find k in the expansion (m ¡ 2n)10 = m10 ¡ 20m9 n + km8 n2 ¡ :::: + 1024n10 .
³
´
q 8
and
10 Find the possible values of q if the constant terms in the expansions of x3 + 3
x
³
´4
q
x3 + 3
are equal.
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896
ANSWERS
EXERCISE 8D
8
10
a 120
b 48
a 48
b 24
12
13
a 15 120
a 3 628 800
c 72
c 15
REVIEW SET 8A
1 a n(n ¡ 1), n > 2
b n+2
2 28
3 a 24
b 6
4 a 900
b 180
5 a a = ex and b = ¡e¡x
b (ex ¡ e¡x )4 = e4x ¡ 4e2x + 6 ¡ 4e¡2x + e¡4x
p
7 It does not have one.
8 c=3
6 362 + 209 3
a 3 628 800 b 241 920
a 360 b 336 c 288
9
11
b 720
b i 151 200
ii 33 600
EXERCISE 8E
9
¡ 13 ¢
4
a
5
a
¡ 12 ¢
b
i
6
= 286
3
5
7
2
9
15
0
a
3
a
¡6¢¡3¢¡7¢
c
¡
1
¡ ¢
2
¡2¢¡8¢
b
0
= 945
2
¡9¢¡7¢
0
¡7¢
x
4
4
= 70
¡ 6 ¢ ¡ 10 ¢
b
2
= 1800
3
8
= 4347
5
EXERCISE 8F
8 2x5 + 11x4 + 24x3 + 26x2 + 14x + 3
a 270
9
¡6¢
2
a 111 +
¡11¢
(2x)1 +
1
b (3x)15 +
14
c (2x)20 +
:::: +
19
a T6 =
c T10 =
a Tr+1 =
5
b 4320
¡ 12 ¢
£26 £(¡3)6
6
x7¡r br
b
2
¡6
¡6¢
1
= 84
10 a = §4
8
a 43 758 teams
b 11 550 teams
10 q = §
p3
c 41 283 teams
35
¢2
+ ::::
¡9¢
3
(x2 )6 y3
¡21¢
8
¡
(2x2 )13 ¡ x1
¢8
(¡3)3
3
b
28 (¡1)4
¡6¢
23 (¡3)3
3
¡9¢
b 2
¡6¢
¡6¢
b b = ¡2
¡9¢
= 28
9 8
+ ::::
¡
3
36 x6 ¡
c
¡6¢
3
23 (¡3)3
¡9¢
4
35 x6 = 91 854x6
(¡2)2 x8 y 8
b n = 6 and k = ¡2
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10 a = 2
75
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r
35 25
2
84x3
75
25
0
5
a
¡7¢
x
(2x)18 ¡ x3
25
6
8 T3 =
9
d T9 =
x
9
¢
2 9
50
a
¢
¡x
¡
8
25
7
¡8¢
2
3 20
¡x
¡
¡ 2 ¢2
¡20¢
b T4 =
¡ ¢
95
d
5
12
4
+
+
0
¡10¢
¡x
¢1
5
a
¢
3 19
95
6
¡ ¢
b 246
(2x)10 + (2x)11
10
(3x)13
2
2 15
x
(2x)10 55
25
5
+
+
a 252
¡11¢
¡15¢
100
a
¡
1
(2x)
x
¡ x3
75
4
¡15¢
¡
19
(2x)
1
¡17¢
100
3
x
¡15¢
5
¡ ¢
14 2 1
¡ 2 ¢14
¡20¢
0
2
(3x)1
¡20¢
(2x)2 + :::: +
2
(3x)
1
¡15¢
:::: +
¡11¢
¡15¢
5
1
2 64:964 808
£ 34 £ (¡2)2 = 4860
9 k = 180
EXERCISE 8G
+x
REVIEW SET 8C
1 a 262 £ 104 = 6 760 000
b 5 £ 26 £ 104 = 1 300 000
c 26 £ 25 £ 10 £ 9 £ 8 £ 7 = 3 276 000
2 a 3003
b 980
c 2982
3 a x3 ¡ 6x2 y + 12xy2 ¡ 8y3
b 81x4 + 216x3 + 216x2 + 96x + 16
4 20 000
5 60
6 k = ¡ 14 , n = 16
7 4320
= 945
2
2 x
n¡1
Hint: Let x = 1 in a.
6 a 9 £ 9 £ 8 £ 7 = 4536 numbers
b 952 numbers
7 (a + b)6 = a6 + 6a5 b + 15a4 b2 + 20a3 b3 + 15a2 b4 + 6ab5 + b6
a x6 ¡ 18x5 + 135x4 ¡ 540x3 + 1215x2 ¡ 1458x + 729
15
6
20
15
6
1
b 1+ + 2 + 3 + 4 + 5 + 6
x
x
x
x
x
x
= 35
3
1
1
¡ n ¢ n¡1 1 ¡ n ¢ n¡2 2 ¡ n ¢ n¡3 3
2
x +
2
x + 3 2
x + ::::
¡ 1 n ¢ 1 n¡1 2 n
REVIEW SET 8B
1 a 45
b 120
= 420
3
= 84
c
2
16
5
¡ 2 ¢ ¡ 10 ¢
ii
b 3n
4 2500
¡ 12 ¢ ¡ 31 ¢ ¡ 7 ¢
0
= 120
:::: +
= 66
2
= 462
6
¡1¢¡9¢
¡ 10 ¢
1
= 792
¡ 2 ¢ ¡ 10 ¢
¡ 3 ¢ ¡ 1 ¢ ¡ 11 ¢
3
¡ 1 ¢ ¡ 12 ¢
b
a 2n +
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